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Theorem bj-2upleq 32193
 Description: Substitution property for ⦅ − , − ⦆. (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-2upleq (𝐴 = 𝐵 → (𝐶 = 𝐷 → ⦅𝐴, 𝐶⦆ = ⦅𝐵, 𝐷⦆))

Proof of Theorem bj-2upleq
StepHypRef Expression
1 bj-1upleq 32180 . . 3 (𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆)
2 bj-xtageq 32169 . . 3 (𝐶 = 𝐷 → ({1𝑜} × tag 𝐶) = ({1𝑜} × tag 𝐷))
3 uneq12 3724 . . . 4 ((⦅𝐴⦆ = ⦅𝐵⦆ ∧ ({1𝑜} × tag 𝐶) = ({1𝑜} × tag 𝐷)) → (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐶)) = (⦅𝐵⦆ ∪ ({1𝑜} × tag 𝐷)))
43ex 449 . . 3 (⦅𝐴⦆ = ⦅𝐵⦆ → (({1𝑜} × tag 𝐶) = ({1𝑜} × tag 𝐷) → (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐶)) = (⦅𝐵⦆ ∪ ({1𝑜} × tag 𝐷))))
51, 2, 4syl2im 39 . 2 (𝐴 = 𝐵 → (𝐶 = 𝐷 → (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐶)) = (⦅𝐵⦆ ∪ ({1𝑜} × tag 𝐷))))
6 df-bj-2upl 32192 . . 3 𝐴, 𝐶⦆ = (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐶))
7 df-bj-2upl 32192 . . 3 𝐵, 𝐷⦆ = (⦅𝐵⦆ ∪ ({1𝑜} × tag 𝐷))
86, 7eqeq12i 2624 . 2 (⦅𝐴, 𝐶⦆ = ⦅𝐵, 𝐷⦆ ↔ (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐶)) = (⦅𝐵⦆ ∪ ({1𝑜} × tag 𝐷)))
95, 8syl6ibr 241 1 (𝐴 = 𝐵 → (𝐶 = 𝐷 → ⦅𝐴, 𝐶⦆ = ⦅𝐵, 𝐷⦆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∪ cun 3538  {csn 4125   × cxp 5036  1𝑜c1o 7440  tag bj-ctag 32155  ⦅bj-c1upl 32178  ⦅bj-c2uple 32191 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-v 3175  df-un 3545  df-opab 4644  df-xp 5044  df-bj-sngl 32147  df-bj-tag 32156  df-bj-1upl 32179  df-bj-2upl 32192 This theorem is referenced by:  bj-2uplth  32202
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